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authorGravatar Alexey Korepanov <kaikaikai@yandex.ru>2012-07-11 16:38:03 -0500
committerGravatar Alexey Korepanov <kaikaikai@yandex.ru>2012-07-11 16:38:03 -0500
commit65db91ac2b544858b54a18b5aad73044753f5650 (patch)
treeff0a1762231cca62ae35df8d23a5aee9a529b6ea
parentba5eecae53aa038374d1708573cf03a2df3f76f3 (diff)
Add a RealQZ class: a generalized Schur decomposition for real matrices
-rw-r--r--Eigen/Eigenvalues1
-rw-r--r--Eigen/src/Eigenvalues/RealQZ.h592
-rw-r--r--test/CMakeLists.txt1
-rw-r--r--test/real_qz.cpp84
4 files changed, 678 insertions, 0 deletions
diff --git a/Eigen/Eigenvalues b/Eigen/Eigenvalues
index af99ccd1f..95c2c462a 100644
--- a/Eigen/Eigenvalues
+++ b/Eigen/Eigenvalues
@@ -27,6 +27,7 @@
#include "src/Eigenvalues/Tridiagonalization.h"
#include "src/Eigenvalues/RealSchur.h"
+#include "src/Eigenvalues/RealQZ.h"
#include "src/Eigenvalues/EigenSolver.h"
#include "src/Eigenvalues/SelfAdjointEigenSolver.h"
#include "src/Eigenvalues/GeneralizedSelfAdjointEigenSolver.h"
diff --git a/Eigen/src/Eigenvalues/RealQZ.h b/Eigen/src/Eigenvalues/RealQZ.h
new file mode 100644
index 000000000..10ab14bc6
--- /dev/null
+++ b/Eigen/src/Eigenvalues/RealQZ.h
@@ -0,0 +1,592 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2012 Alexey Korepanov <kaikaikai@yandex.ru>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+/* TODO:
+ * moar documentation
+ *
+ * Scalar(0), Scalar(0.5), etc
+ * use coeffRef?
+ *
+ */
+
+#ifndef EIGEN_REAL_QZ_H
+#define EIGEN_REAL_QZ_H
+
+namespace Eigen {
+
+ /** \eigenvalues_module \ingroup Eigenvalues_Module
+ *
+ *
+ * \class RealQZ
+ *
+ * \brief Performs a real QZ decomposition of a pair of square matrices
+ *
+ * \tparam _MatrixType the type of the matrix of which we are computing the
+ * real QZ decomposition; this is expected to be an instantiation of the
+ * Matrix class template.
+ *
+ * Given a real square matrices A and B, this class computes the real QZ
+ * decomposition: \f$ A = Q S Z \f$, \f$ B = Q T Z \f$ where Q and Z are
+ * real orthogonal matrixes, T is upper-triangular matrix, and S is upper
+ * quasi-triangular matrix. An orthogonal matrix is a matrix whose
+ * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
+ * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
+ * blocks and 2-by-2 blocks where further reduction is impossible due to
+ * complex eigenvalues.
+ *
+ * The eigenvalues of the pencil \f$ A - z B \f$ can be obtained from
+ * 1x1 and 2x2 blocks on the diagonals of S and T.
+ *
+ * Call the function compute() to compute the real QZ decomposition of a
+ * given pair of matrices. Alternatively, you can use the
+ * RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ)
+ * constructor which computes the real QZ decomposition at construction
+ * time. Once the decomposition is computed, you can use the matrixS(),
+ * matrixT(), matrixQ() and matrixZ() functions to retrieve the matrices
+ * S, T, Q and Z in the decomposition. If computeQZ==false, some time
+ * is saved by not computing matrices Q and Z.
+ *
+ * I should add an example of usage of this class, but I don't exactly know
+ * how.
+ *
+ * \note The implementation is based on the algorithm in "Matrix Computations"
+ * by Gene H. Golub and Charles F. Van Loan, and a paper "An algorithm for
+ * generalized eigenvalue problems" by C.B.Moler and G.W.Stewart.
+ *
+ * \sa class RealSchur, class ComplexSchur, class EigenSolver, class ComplexEigenSolver
+ */
+
+ template<typename _MatrixType> class RealQZ {
+ public:
+ typedef _MatrixType MatrixType;
+ enum {
+ RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+ ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+ Options = MatrixType::Options,
+ MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+ MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
+ };
+ typedef typename MatrixType::Scalar Scalar;
+ typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
+ typedef typename MatrixType::Index Index;
+
+ typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
+ typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
+
+ /** \brief Default constructor.
+ *
+ * \param [in] size Positive integer, size of the matrix whose QZ decomposition will be computed.
+ *
+ * The default constructor is useful in cases in which the user intends to
+ * perform decompositions via compute(). The \p size parameter is only
+ * used as a hint. It is not an error to give a wrong \p size, but it may
+ * impair performance.
+ *
+ * \sa compute() for an example.
+ */
+ RealQZ(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) :
+ m_S(size, size),
+ m_T(size, size),
+ m_Q(size, size),
+ m_Z(size, size),
+ m_workspace(size*2),
+ m_isInitialized(false)
+ { }
+
+ /** \brief Constructor; computes real QZ decomposition of given matrices
+ *
+ * \param[in] A Matrix A.
+ * \param[in] B Matrix B.
+ * \param[in] computeQZ If false, A and Z are not computed.
+ *
+ * This constructor calls compute() to compute the QZ decomposition.
+ */
+ RealQZ(const MatrixType& A, const MatrixType& B, bool computeQZ = true) :
+ m_S(A.rows(),A.cols()),
+ m_T(A.rows(),A.cols()),
+ m_Q(A.rows(),A.cols()),
+ m_Z(A.rows(),A.cols()),
+ m_workspace(A.rows()*2),
+ m_isInitialized(false) {
+ compute(A, B, computeQZ);
+ }
+
+ /** \brief Returns matrix Q in the QZ decomposition.
+ *
+ * \returns A const reference to the matrix Q.
+ */
+ const MatrixType& matrixQ() const {
+ eigen_assert(m_isInitialized && "RealQZ is not initialized.");
+ eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition.");
+ return m_Q;
+ }
+
+ /** \brief Returns matrix Z in the QZ decomposition.
+ *
+ * \returns A const reference to the matrix Z.
+ */
+ const MatrixType& matrixZ() const {
+ eigen_assert(m_isInitialized && "RealQZ is not initialized.");
+ eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition.");
+ return m_Z;
+ }
+
+ /** \brief Returns matrix S in the QZ decomposition.
+ *
+ * \returns A const reference to the matrix S.
+ */
+ const MatrixType& matrixS() const {
+ eigen_assert(m_isInitialized && "RealQZ is not initialized.");
+ return m_S;
+ }
+
+ /** \brief Returns matrix S in the QZ decomposition.
+ *
+ * \returns A const reference to the matrix S.
+ */
+ const MatrixType& matrixT() const {
+ eigen_assert(m_isInitialized && "RealQZ is not initialized.");
+ return m_T;
+ }
+
+ /** \brief Computes QZ decomposition of given matrix.
+ *
+ * \param[in] A Matrix A.
+ * \param[in] B Matrix B.
+ * \param[in] computeQZ If false, A and Z are not computed.
+ * \returns Reference to \c *this
+ */
+ RealQZ& compute(const MatrixType& A, const MatrixType& B, bool computeQZ = true);
+
+ /** \brief Reports whether previous computation was successful.
+ *
+ * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
+ */
+ ComputationInfo info() const
+ {
+ eigen_assert(m_isInitialized && "RealQZ is not initialized.");
+ return m_info;
+ }
+
+ /** \brief Returns number of performed QR-like iterations.
+ */
+ Index iterations() const
+ {
+ eigen_assert(m_isInitialized && "RealQZ is not initialized.");
+ return m_global_iter;
+ }
+
+ /** \brief Maximum number of iterations.
+ *
+ * Maximum number of iterations allowed for an eigenvalue to converge.
+ */
+ static const Index m_max_iter = 400;
+
+ private:
+
+ MatrixType m_S, m_T, m_Q, m_Z;
+ Matrix<Scalar,Dynamic,1> m_workspace;
+ ComputationInfo m_info;
+ bool m_isInitialized;
+ bool m_computeQZ;
+ Scalar m_normOfT, m_normOfS;
+ Index m_global_iter;
+
+ typedef Matrix<Scalar,3,1> Vector3s;
+ typedef Matrix<Scalar,2,1> Vector2s;
+ typedef Matrix<Scalar,2,2> Matrix2s;
+ typedef JacobiRotation<Scalar> JRs;
+
+ void hessenbergTriangular();
+ void computeNorms();
+ Index findSmallSubdiagEntry(Index iu);
+ Index findSmallDiagEntry(Index f, Index l);
+ void splitOffTwoRows(Index i);
+ void pushDownZero(Index z, Index f, Index l);
+ void step(Index f, Index l, Index iter);
+
+ }; // RealQZ
+
+ /** \internal Reduces S and T to upper Hessenberg - triangular form */
+ template<typename MatrixType>
+ void RealQZ<MatrixType>::hessenbergTriangular() {
+
+ const Index dim = m_S.cols();
+
+ // perform QR decomposition of T, overwrite T with R, save Q
+ HouseholderQR<MatrixType> qrT(m_T);
+ m_T = qrT.matrixQR();
+ m_T.template triangularView<StrictlyLower>().setZero();
+ m_Q = qrT.householderQ();
+ // overwrite S with Q* S
+ m_S.applyOnTheLeft(m_Q.adjoint());
+ // init Z as Identity
+ if (m_computeQZ)
+ m_Z = MatrixType::Identity(dim,dim);
+ // reduce S to upper Hessenberg with Givens rotations
+ for (Index j=0; j<=dim-3; j++) {
+ for (Index i=dim-1; i>=j+2; i--) {
+ JRs G;
+ // kill S(i,j)
+ G.makeGivens(m_S.coeff(i-1, j), m_S.coeff(i,j));
+ m_S.applyOnTheLeft(i-1,i,G.adjoint());
+ m_T.applyOnTheLeft(i-1,i,G.adjoint());
+ m_S.coeffRef(i,j) = Scalar(0.0);
+ // update Q
+ if (m_computeQZ)
+ m_Q.applyOnTheRight(i-1,i,G);
+ // kill T(i,i-1)
+ G.makeGivens(m_T.coeff(i,i), m_T.coeff(i,i-1));
+ m_S.applyOnTheRight(i,i-1,G);
+ m_T.applyOnTheRight(i,i-1,G);
+ m_T.coeffRef(i,i-1) = Scalar(0.0);
+ // update Z
+ if (m_computeQZ)
+ m_Z.applyOnTheLeft(i,i-1,G.adjoint());
+ }
+ }
+ }
+
+ /** \internal Computes vector L1 norms of S and T when in Hessenberg-Triangular form already */
+ template<typename MatrixType>
+ inline void RealQZ<MatrixType>::computeNorms() {
+ const Index size = m_S.cols();
+ m_normOfS = Scalar(0.0);
+ m_normOfT = Scalar(0.0);
+ for (Index j = 0; j < size; ++j) {
+ Index row_start = (std::max)(j-1,Index(0));
+ m_normOfS += m_S.row(j).segment(row_start, size - row_start).cwiseAbs().sum();
+ m_normOfT += m_T.row(j).segment(j, size - j).cwiseAbs().sum();
+ }
+ }
+
+
+ /** \internal Look for single small sub-diagonal element S(res, res-1) and return res (or 0) */
+ template<typename MatrixType>
+ inline typename MatrixType::Index RealQZ<MatrixType>::findSmallSubdiagEntry(Index iu) {
+ Index res = iu;
+ while (res > 0) {
+ Scalar s = internal::abs(m_S.coeff(res-1,res-1)) + internal::abs(m_S.coeff(res,res));
+ if (s == Scalar(0.0))
+ s = m_normOfS;
+ if (internal::abs(m_S.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
+ break;
+ res--;
+ }
+ return res;
+ }
+
+ /** \internal Look for single small diagonal element T(res, res) for res between f and l, and return res (or f-1) */
+ template<typename MatrixType>
+ inline typename MatrixType::Index RealQZ<MatrixType>::findSmallDiagEntry(Index f, Index l) {
+ Index res = l;
+ while (res >= f) {
+ if (internal::abs(m_T.coeff(res,res)) <= NumTraits<Scalar>::epsilon() * m_normOfT)
+ break;
+ res--;
+ }
+ return res;
+ }
+
+ /** \internal decouple 2x2 diagonal block in rows iu, iu+1 if eigenvalues are real */
+ template<typename MatrixType>
+ inline void RealQZ<MatrixType>::splitOffTwoRows(Index i) {
+ const Index dim=m_S.cols();
+ if (internal::abs(m_S.coeff(i+1,1)==Scalar(0)))
+ return;
+ Index z = findSmallDiagEntry(i,i+1);
+ if (z==i-1) {
+ // block of (S T^{-1})
+ Matrix2s STi = m_T.template block<2,2>(i,i).template triangularView<Upper>().
+ template solve<OnTheRight>(m_S.template block<2,2>(i,i));
+ Scalar p = Scalar(0.5)*(STi(0,0)-STi(1,1));
+ Scalar q = p*p + STi(1,0)*STi(0,1);
+ if (q>=0) {
+ Scalar z = internal::sqrt(q);
+ // QR for ABi - lambda I
+ JRs G;
+ if (p>=0)
+ G.makeGivens(p + z, STi(1,0));
+ else
+ G.makeGivens(p - z, STi(1,0));
+ m_S.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());
+ m_T.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());
+ // update Q
+ if (m_computeQZ)
+ m_Q.applyOnTheRight(i,i+1,G);
+
+ G.makeGivens(m_T.coeff(i+1,i+1), m_T.coeff(i+1,i));
+ m_S.topRows(i+2).applyOnTheRight(i+1,i,G);
+ m_T.topRows(i+2).applyOnTheRight(i+1,i,G);
+ // update Z
+ if (m_computeQZ)
+ m_Z.applyOnTheLeft(i+1,i,G.adjoint());
+
+ m_S.coeffRef(i+1,i) = Scalar(0.0);
+ m_T.coeffRef(i+1,i) = Scalar(0.0);
+ }
+ } else {
+ pushDownZero(z,i,i+1);
+ }
+ }
+
+ /** \internal use zero in T(z,z) to zero S(l,l-1), working in block f..l */
+ template<typename MatrixType>
+ inline void RealQZ<MatrixType>::pushDownZero(Index z, Index f, Index l) {
+ JRs G;
+ const Index dim = m_S.cols();
+ for (Index zz=z; zz<l; zz++) {
+ // push 0 down
+ Index firstColS = zz>f ? (zz-1) : zz;
+ G.makeGivens(m_T.coeff(zz, zz+1), m_T.coeff(zz+1, zz+1));
+ m_S.rightCols(dim-firstColS).applyOnTheLeft(zz,zz+1,G.adjoint());
+ m_T.rightCols(dim-zz).applyOnTheLeft(zz,zz+1,G.adjoint());
+ m_T.coeffRef(zz+1,zz+1) = Scalar(0.0);
+ // update Q
+ if (m_computeQZ)
+ m_Q.applyOnTheRight(zz,zz+1,G);
+ // kill S(zz+1, zz-1)
+ if (zz>f) {
+ G.makeGivens(m_S.coeff(zz+1, zz), m_S.coeff(zz+1,zz-1));
+ m_S.bottomRows(dim-zz).applyOnTheRight(zz, zz-1,G);
+ m_T.bottomRows(dim-zz).applyOnTheRight(zz, zz-1,G);
+ m_S.coeffRef(zz+1,zz-1) = Scalar(0.0);
+ // update Z
+ if (m_computeQZ)
+ m_Z.applyOnTheLeft(zz,zz-1,G.adjoint());
+ }
+ }
+ // finally kill S(l,l-1)
+ G.makeGivens(m_S.coeff(l,l), m_S.coeff(l,l-1));
+ m_S.applyOnTheRight(l,l-1,G);
+ m_T.applyOnTheRight(l,l-1,G);
+ m_S.coeffRef(l,l-1)=Scalar(0.0);
+ // update Z
+ if (m_computeQZ)
+ m_Z.applyOnTheLeft(l,l-1,G.adjoint());
+ }
+
+ /** \internal QR-like iterative step */
+ template<typename MatrixType>
+ inline void RealQZ<MatrixType>::step(Index f, Index l, Index iter) {
+ const Index dim = m_S.cols();
+
+ // x, y, z
+ Scalar x, y, z;
+ if (iter==10) {
+ // Wilkinson ad hoc shift
+ const Scalar
+ a11=m_S.coeff(f+0,f+0), a12=m_S.coeff(f+0,f+1),
+ a21=m_S.coeff(f+1,f+0), a22=m_S.coeff(f+1,f+1), a32=m_S.coeff(f+2,f+1),
+ b12=m_T.coeff(f+0,f+1),
+ b11i=Scalar(1.0)/m_T.coeff(f+0,f+0),
+ b22i=Scalar(1.0)/m_T.coeff(f+1,f+1),
+ a87=m_S.coeff(l-1,l-2),
+ a98=m_S.coeff(l-0,l-1),
+ b77i=Scalar(1.0)/m_T.coeff(l-2,l-2),
+ b88i=Scalar(1.0)/m_T.coeff(l-1,l-1);
+ Scalar ss = internal::abs(a87*b77i) + internal::abs(a98*b88i),
+ lpl = Scalar(1.5)*ss,
+ ll = ss*ss;
+ x = ll + a11*a11*b11i*b11i - lpl*a11*b11i + a12*a21*b11i*b22i
+ - a11*a21*b12*b11i*b11i*b22i;
+ y = a11*a21*b11i*b11i - lpl*a21*b11i + a21*a22*b11i*b22i
+ - a21*a21*b12*b11i*b11i*b22i;
+ z = a21*a32*b11i*b22i;
+ } else if (iter==16) {
+ // another exceptional shift
+ x = m_S.coeff(f,f)/m_T.coeff(f,f)-m_S.coeff(l,l)/m_T.coeff(l,l) + m_S.coeff(l,l-1)*m_T.coeff(l-1,l) /
+ (m_T.coeff(l-1,l-1)*m_T.coeff(l,l));
+ y = m_S.coeff(f+1,f)/m_T.coeff(f,f);
+ z = 0;
+ } else if (iter>23 && !(iter%8)) {
+ // extremely exceptional shift
+ x = internal::random<Scalar>(-1.0,1.0);
+ y = internal::random<Scalar>(-1.0,1.0);
+ z = internal::random<Scalar>(-1.0,1.0);
+ } else {
+ const Scalar
+ a11=m_S.coeff(f,f), a12=m_S.coeff(f,f+1),
+ a21=m_S.coeff(f+1,f), a22=m_S.coeff(f+1,f+1),
+ a32=m_S.coeff(f+2,f+1),
+ a88=m_S.coeff(l-1,l-1), a89=m_S.coeff(l-1,l),
+ a98=m_S.coeff(l,l-1), a99=m_S.coeff(l,l),
+ b11=m_T.coeff(f,f), b11i=1.0/b11, b12=m_T.coeff(f,f+1),
+ b22i=Scalar(1.0)/m_T.coeff(f+1,f+1),
+ b88i=Scalar(1.0)/m_T.coeff(l-1,l-1), b89=m_T.coeff(l-1,l),
+ b99i=Scalar(1.0)/m_T.coeff(l,l);
+ x = ( (a88*b88i - a11*b11i)*(a99*b99i - a11*b11i) - (a89*b99i)*(a98*b88i) + (a98*b88i)*(b89*b99i)*(a11*b11i) ) * (b11/a21)
+ + a12*b22i - (a11*b11i)*(b12*b22i);
+ y = (a22*b22i-a11*b11i) - (a21*b11i)*(b12*b22i) - (a88*b88i-a11*b11i) - (a99*b99i-a11*b11i) + (a98*b88i)*(b89*b99i);
+ z = a32*b22i;
+ }
+
+ JRs G;
+
+ for (Index k=f; k<=l-2; k++) {
+ // variables for Householder reflections
+ Vector2s essential2;
+ Scalar tau, beta;
+
+ Vector3s hr(x,y,z);
+
+ // Q_k
+ hr.makeHouseholderInPlace(tau, beta);
+ essential2 = hr.template bottomRows<2>();
+ Index fc=(std::max)(k-1,Index(0)); // first col to update
+ m_S.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());
+ m_T.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());
+ if (m_computeQZ)
+ m_Q.template middleCols<3>(k).applyHouseholderOnTheRight(essential2, tau, m_workspace.data());
+ if (k>f) {
+ m_S.coeffRef(k+1,k-1) = Scalar(0.0);
+ m_S.coeffRef(k+2,k-1) = Scalar(0.0);
+ }
+
+ // Z_{k1}
+ hr << m_T.coeff(k+2,k+2),m_T.coeff(k+2,k),m_T.coeff(k+2,k+1);
+ hr.makeHouseholderInPlace(tau, beta);
+ essential2 = hr.template bottomRows<2>();
+ {
+ Index lr = (std::min)(k+4,dim); // last row to update
+ Map<Matrix<Scalar,Dynamic,1> > tmp(m_workspace.data(),lr);
+ // S
+ tmp = m_S.template middleCols<2>(k).topRows(lr) * essential2;
+ tmp += m_S.col(k+2).head(lr);
+ m_S.col(k+2).head(lr) -= tau*tmp;
+ m_S.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();
+ // T
+ tmp = m_T.template middleCols<2>(k).topRows(lr) * essential2;
+ tmp += m_T.col(k+2).head(lr);
+ m_T.col(k+2).head(lr) -= tau*tmp;
+ m_T.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();
+ }
+ if (m_computeQZ) {
+ // Z
+ Map<Matrix<Scalar,1,Dynamic> > tmp(m_workspace.data(),dim);
+ tmp = essential2.adjoint()*(m_Z.template middleRows<2>(k));
+ tmp += m_Z.row(k+2);
+ m_Z.row(k+2) -= tau*tmp;
+ m_Z.template middleRows<2>(k) -= essential2 * (tau*tmp);
+ }
+ m_T.coeffRef(k+2,k) = Scalar(0.0);
+ m_T.coeffRef(k+2,k+1) = Scalar(0.0);
+
+ // Z_{k2}
+ G.makeGivens(m_T.coeff(k+1,k+1), m_T.coeff(k+1,k));
+ m_S.applyOnTheRight(k+1,k,G);
+ m_T.applyOnTheRight(k+1,k,G);
+ // update Z
+ if (m_computeQZ)
+ m_Z.applyOnTheLeft(k+1,k,G.adjoint());
+ m_T.coeffRef(k+1,k) = Scalar(0.0);
+
+ // update x,y,z
+ x = m_S.coeff(k+1,k);
+ y = m_S.coeff(k+2,k);
+ if (k < l-2)
+ z = m_S.coeff(k+3,k);
+ } // loop over k
+
+ // Q_{n-1}
+ G.makeGivens(x,y);
+ m_S.applyOnTheLeft(l-1,l,G.adjoint());
+ m_T.applyOnTheLeft(l-1,l,G.adjoint());
+ if (m_computeQZ)
+ m_Q.applyOnTheRight(l-1,l,G);
+ m_S.coeffRef(l,l-2) = Scalar(0.0);
+
+ // Z_{n-1}
+ G.makeGivens(m_T.coeff(l,l),m_T.coeff(l,l-1));
+ m_S.applyOnTheRight(l,l-1,G);
+ m_T.applyOnTheRight(l,l-1,G);
+ if (m_computeQZ)
+ m_Z.applyOnTheLeft(l,l-1,G.adjoint());
+ m_T.coeffRef(l,l-1) = Scalar(0.0);
+
+ }
+
+
+ template<typename MatrixType>
+ RealQZ<MatrixType>& RealQZ<MatrixType>::compute(const MatrixType& A_in, const MatrixType& B_in, bool computeQZ) {
+
+ const Index dim = A_in.cols();
+
+ assert (A_in.rows()==dim && A_in.cols()==dim
+ && B_in.rows()==dim && B_in.cols()==dim
+ && "Need square matrices of the same dimension");
+
+ m_isInitialized = true;
+ m_computeQZ = computeQZ;
+ m_S = A_in; m_T = B_in;
+ m_workspace.resize(dim*2);
+ m_global_iter = 0;
+
+ // entrance point: hessenberg triangular decomposition
+ hessenbergTriangular();
+ // compute L1 vector norms of T, S into m_normOfS, m_normOfT
+ computeNorms();
+
+ Index l = dim-1,
+ f,
+ local_iter = 0;
+
+ while (l>0 && local_iter<m_max_iter) {
+ f = findSmallSubdiagEntry(l);
+ if (f>0) m_S.coeffRef(f,f-1) = Scalar(0.0);
+ if (f == l) {
+ l --;
+ local_iter = 0;
+ } else if (f == l-1) {
+ splitOffTwoRows(f);
+ l -= 2;
+ local_iter = 0;
+ } else {
+ Index z = findSmallDiagEntry(f,l);
+ if (z>=f) {
+ // zero found
+ pushDownZero(z,f,l);
+ } else {
+ // QR-like iteration
+ step(f,l, local_iter);
+ local_iter++;
+ m_global_iter++;
+ }
+ }
+ }
+ // check if we converged before reaching iterations limit
+ if (local_iter<m_max_iter) {
+ m_info = Success;
+ } else {
+ m_info = NoConvergence;
+ }
+ return *this;
+ } // end compute
+
+
+} // end namespace Eigen
+
+
+#endif //EIGEN_REAL_QZ
+
diff --git a/test/CMakeLists.txt b/test/CMakeLists.txt
index c71534df9..69250fb0a 100644
--- a/test/CMakeLists.txt
+++ b/test/CMakeLists.txt
@@ -155,6 +155,7 @@ ei_add_test(inverse)
ei_add_test(qr)
ei_add_test(qr_colpivoting)
ei_add_test(qr_fullpivoting)
+ei_add_test(real_qz)
ei_add_test(upperbidiagonalization)
ei_add_test(hessenberg)
ei_add_test(schur_real)
diff --git a/test/real_qz.cpp b/test/real_qz.cpp
new file mode 100644
index 000000000..e5b301c94
--- /dev/null
+++ b/test/real_qz.cpp
@@ -0,0 +1,84 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2012 Alexey Korepanov <kaikaikai@yandex.ru>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#include "main.h"
+#include <limits>
+#include <Eigen/Eigenvalues>
+
+template<typename MatrixType> void real_qz(const MatrixType& m)
+{
+ /* this test covers the following files:
+ RealQZ.h
+ */
+
+ typedef typename MatrixType::Index Index;
+ typedef typename MatrixType::Scalar Scalar;
+ typedef typename NumTraits<Scalar>::Real RealScalar;
+ typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
+ typedef Matrix<RealScalar, MatrixType::RowsAtCompileTime, 1> RealVectorType;
+ typedef typename std::complex<typename NumTraits<typename MatrixType::Scalar>::Real> Complex;
+
+ Index dim = m.cols();
+
+ MatrixType A = MatrixType::Random(dim,dim),
+ B = MatrixType::Random(dim,dim);
+
+ RealQZ<MatrixType> qz(A,B);
+
+ VERIFY_IS_EQUAL(qz.info(), Success);
+ // check for zeros
+ bool all_zeros = true;
+ for (Index i=0; i<A.cols(); i++)
+ for (Index j=0; j<i; j++) {
+ if (internal::abs(qz.matrixT()(i,j))!=Scalar(0.0))
+ all_zeros = false;
+ if (j<i-1 && internal::abs(qz.matrixS()(i,j))!=Scalar(0.0))
+ all_zeros = false;
+ if (j==i-1 && j>0 && internal::abs(qz.matrixS()(i,j))!=Scalar(0.0) && internal::abs(qz.matrixS()(i-1,j-1))!=Scalar(0.0))
+ all_zeros = false;
+ }
+ VERIFY_IS_EQUAL(all_zeros, true);
+ VERIFY_IS_APPROX(qz.matrixQ()*qz.matrixS()*qz.matrixZ(), A);
+ VERIFY_IS_APPROX(qz.matrixQ()*qz.matrixT()*qz.matrixZ(), B);
+ VERIFY_IS_APPROX(qz.matrixQ()*qz.matrixQ().adjoint(), MatrixType::Identity(dim,dim));
+ VERIFY_IS_APPROX(qz.matrixZ()*qz.matrixZ().adjoint(), MatrixType::Identity(dim,dim));
+}
+
+void test_real_qz()
+{
+ int s;
+ for(int i = 0; i < g_repeat; i++) {
+ CALL_SUBTEST_1( real_qz(Matrix4f()) );
+ s = internal::random<int>(1,EIGEN_TEST_MAX_SIZE/4);
+ CALL_SUBTEST_2( real_qz(MatrixXd(s,s)) );
+
+ // some trivial but implementation-wise tricky cases
+ CALL_SUBTEST_2( real_qz(MatrixXd(1,1)) );
+ CALL_SUBTEST_2( real_qz(MatrixXd(2,2)) );
+ CALL_SUBTEST_3( real_qz(Matrix<double,1,1>()) );
+ CALL_SUBTEST_4( real_qz(Matrix2d()) );
+ }
+
+ EIGEN_UNUSED_VARIABLE(s)
+}